IT-t Mr. P. Nicholson on the Ttansfunnat ion of Functions. 



This mode of operation will be observed in the following 

 examples. 



Ex.2. Transform the function ^' — 9jr'4-27a:' — ^Lr+SO 

 into another which shall have x— 4 instead of .r. 



Opcratio7i. 



1—9 + 27-41 + 30( 4 



— 5 



-1+ 7 

 +3+ 3-13 



l-f7+15- 1-22 



And thus the function j:*— 9jr' + 27a'- — 4U' + 30 is trans- 

 formed to (.r-4.)'+7(.r— 4)'-f 15(.r-4)'— (j:-4)-22. 



Ex: 3. Transform the function .r^ — 3x' — 3x*-|-15.r — 29 

 into another function which shall have x-\-'3 instead of .r. 



Opoation. 

 1 — 3- 3+- 15-29( - 3 



— 9+15 



— 12+42— 30 



— 15+78-156+61 



So that the original function .r*— 3.r' — 3.i-^+15j — 29 is trans- 

 formed to (.r+3V-15(x+3)'+78(.r+3)^-156(x+3)+61. 



JEr. 4. Transform the function a' — 7x^ + 17cr— 15 into 

 another function which shall have .r — J instead of .r. 



In this example it will be the most eligible to find equiva- 

 lents to the coefficients and absolute number in the form of 

 fractions, so that the denominators may be the regular powers 

 of the denominator of the fraction, by which the base x of the 

 original function is to be diminished ; and then perform the 

 operation with the numerators instead of the respective co- 

 efficients, as if whole numbers, then .r by 7. 



XT « 3'^ 21 ,^ 3'-I7 153 , y-\5 -105 



Now 7=— = -, 17=-|p-=— , and 15=^3-= „^. 



Whence the operation 



1-21+153-405(7 



— 14 



— 7+55 



1+ 0+ 6 — 20 

 The original function j:' — 7^*+1 7.i — 1 5 is thus transformed 

 to(x-5)'±0+§(.r-i)-ff. _ 



So that in lact this process is equivalent to taknig away the 

 second term, as it is called, or to transform the equation to 

 another which shall want the second term. 



Ex. 



